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 I suspect more variance in the partners in more difficult environments, perhaps? Europe and West Africa have been particularly difficult places. 
 Very possible. 

But the effect is in an unexpected direction - more monopolising of mating opportunities by high-status males, at least in the form of high remarriage rates.



 
 Yes, that's what I meant.
In the West, at least, men are more likely to remarry after divorce, which would increase the number of men with children from more than one woman.
Western women are also probably more likely to have children from more than one man.